An asymptotic formula for the t-core partition function and a conjecture of Stanton

For a positive integer t, a partition is said to be a t-core if each of the hook numbers from its Ferrers–Young diagram is not a multiple of t. In 1996, Granville and Ono proved the t-core partition conjecture, that at(n), the number of t-core partitions of n, is positive for every nonnegative integer n as long as t⩾4. As part of their proof, they showed that if p⩾5 is prime, the generating function for ap(n) is essentially a multiple of an explicit Eisenstein Series together with a cusp form. This representation of the generating function leads to an asymptotic formula for ap(n) involving L-functions and divisor functions. In 1999, Stanton conjectured that for t⩾4 and n⩾t+1, at(n)⩽at+1(n). Here we prove a weaker form of this conjecture, that for t⩾4 and n sufficiently large, at(n)⩽at+1(n). Along the way, we obtain an asymptotic formula for at(n) which, in the cases where t is coprime to 6, is a generalization of the formula which follows from the work of Granville and Ono when t=p⩾5 is prime.